1153
J. Synchrotron Rad. (1999). 6, 1153±1167
Imaging by parabolic refractive lenses in the hard X-ray range
Bruno Lengeler,a* Christian Schroer,a Johannes TuÈmmler,a Boris Benner,a
Matthias Richwin,a Anatoly Snigirev,b Irina Snigirevab and
Michael Drakopoulosb
a
Zweites Physikalisches Institut, RWTH Aachen, D-52056 Aachen, Germany, and bEuropean
Synchrotron Radiation Facility (ESRF), BP 220, F-38043 Grenoble CEDEX, France.
E-mail: [email protected]
(Received 20 April 1999; accepted 15 July 1999 )
The manufacture and properties of compound refractive lenses (CRLs) for hard X-rays with
parabolic pro®le are described. These novel lenses can be used up to 60 keV. A typical focal length
is 1 m. They have a geometrical aperture of 1 mm and are best adapted to undulator beams at
synchrotron radiation sources. The transmission ranges from a few % in aluminium CRLs up to about
30% expected in beryllium CRLs. The gain (ratio of the intensity in the focal spot relative to the
intensity behind a pinhole of equal size) is larger than 100 for aluminium and larger than 1000 for
beryllium CRLs. Due to their parabolic pro®le they are free of spherical aberration and are genuine
imaging devices. The theory for imaging an X-ray source and an object illuminated by it has been
developed, including the effects of attenuation (photoabsorption and Compton scattering) and of the
roughness at the lens surface. Excellent agreement between theory and experiment has been found.
With aluminium CRLs a lateral resolution in imaging of 0.3 mm has been achieved and a resolution
below 0.1 mm can be expected for beryllium CRLs. The main ®elds of application of the refractive
X-ray lenses are (i) microanalysis with a beam in the micrometre range for diffraction, ¯uorescence,
absorption, scattering; (ii) imaging in absorption and phase contrast of opaque objects which cannot
tolerate sample preparation; (iii) coherent X-ray scattering.
Keywords: X-ray optics; microanalysis; X-ray imaging; coherent X-ray scattering.
1. Introduction
Microanalysis and imaging with X-rays have developed
rapidly over the last few years. They are a major part of the
activities at synchrotron radiation sources. Microdiffraction, micro¯uorescence, microabsorption spectroscopy and
X-ray microscopy in absorption and phase contrast have
many applications in basic science and technology. While
microanalysis requires devices which generate a small focal
spot, X-ray microscopy calls for high-quality imaging
components. Up to now, curved mirrors (Kirkpatrick &
Baez, 1948; Suzuki & Uchida, 1992) and multilayers
(Underwood et al., 1986, 1988), single and multiple capillaries (Bilderback et al., 1994; Hoffman et al., 1994) and
diffracting lenses (Aristov et al., 1986; Lai et al., 1992;
Snigirev, 1995; Snigireva et al., 1998) are the standard
means of generating a small focal spot. Some of these
devices are suited for imaging (Underwood, 1986; Tarazona
et al., 1994; Hartman et al., 1995; Snigirev et al., 1997; Yun et
al., 1998). Recently, we have shown that there are means of
manufacturing refractive X-ray lenses. Refractive lenses
were considered for a long time not to be feasible at all, or
at least dif®cult to fabricate and inferior in quality to
Fresnel lenses (Yang, 1993). The ®rst lenses of this type had
# 1999 International Union of Crystallography
Printed in Great Britain ± all rights reserved
cylinder and crossed-cylinder symmetry with a focal length
in the metre range and a focal spot size in the micrometre
range (Snigirev et al., 1996; Snigirev, Filseth et al., 1998;
Elleaume, 1998). A large number of holes, 0.5±1 mm in
diameter, are drilled in aluminium or beryllium to form a
well aligned row of holes. The material between the holes
focuses the beam. Due to the assembly of a large number of
individual lenses these refractive X-ray lenses are called
compound refractive lenses (CRLs). The concept has also
been transferred to the focusing of neutron beams
(Eskildsen et al., 1998). However, these lenses show strong
spherical aberration and are not well suited for imaging
purposes due to imperfections in the lens shaping.
In the meantime, we have considerably improved the
concept and the manufacturing of the CRL. The new lenses
have a parabolic pro®le and rotational symmetry around
the optical axis. Hence, they focus in two directions and are
free of spherical aberration. They are genuine imaging
devices, like glass lenses for visible light. They can withstand the full radiation (white beam) of an undulator at
third-generation synchrotron radiation sources. The lenses
are mechanically robust and easy to align and to operate.
The paper is organized as follows. x2 gives details on the
design and on the manufacturing of parabolic CRLs. In x3
Journal of Synchrotron Radiation
ISSN 0909-0495 # 1999
1154
Parabolic refractive lenses
Table 1
Typical values of and for beryllium and aluminium as
refractive lens material.
Values of X-ray attenuation (photoabsorption and Compton scattering)
and of Z ‡ f 0 from Henke et al. (1993).
E (keV)
(10ÿ6 )
8
10
15
20
30
5.334
3.412
1.515
0.852
0.379
Be
(10ÿ9 )
(10ÿ6 )
2.419
1.095
0.341
0.195
0.113
8.579
5.468
2.414
1.355
0.601
Al
(10ÿ9 )
158.20
66.61
13.50
4.40
0.95
we consider the imaging of an undulator source by a CRL.
x4 deals with the imaging of an object illuminated by an
undulator beam by means of a CRL. A comparison is made
between theory and experimental results. Finally, in x5 we
give a summary and an outlook on further developments
and on applications of the new parabolic CRL.
2. Refractive X-ray lenses
It has been textbook knowledge for many years that there
are no refractive lenses for X-rays (Lipson et al., 1998). The
lens-maker formula for a convex lens with equal radii of
curvature R on both sides reads as
1=f ˆ 2…n ÿ 1†=R;
lens material (Lengeler et al., 1998; Snigirev, Filseth et al.,
1998; Snigirev, Kohn et al., 1998). If, in addition, the pro®le
of the lens is a paraboloid of rotation, spherical aberration
can be eliminated (Lengeler et al., 1998, 1999). Under these
conditions a focal length of 1 m and a transmission between
1 and 30% can be achieved as shown in this paper. Table 1
shows some typical values of and for materials which
are good candidates for refractive X-ray lenses. It turns out
that for all energies the value is much smaller than the value.
In Appendix A it is shown that the focal length of a CRL
with parabolic pro®le s2 ˆ 2Rw (Fig. 1) is
f ˆ …R=2N†‰1 ‡ O…†Š;
…5†
as measured from the middle of the lens. R is the radius of
curvature at the apex of the parabolas, N is the number of
individual lenses in the stack. A lens with a thickness
2w0 ‡ d has an aperture 2R0 ˆ 2…2Rw0 †1=2 (Fig. 1). For
typical values quoted below, in particular for f ' 1 m, the
correction term of order in expression (5) is less than
1 mm and will be neglected. Let us consider a typical
example. With tabulated f 0 values (Henke et al., 1993),
ˆ 2:41 10ÿ6 for aluminium at 15 keV. For a radius of
curvature R ˆ 0:2 mm we obtain f ˆ 0:99 m if N ˆ 42.
With 2w0 ‡ d ˆ 1 mm the lens will have a length of 42 mm,
…1†
where n is the index of refraction. For visible light in glass,
absorption can be neglected and n is 1.5. In that case a
radius of curvature of 10 cm gives a focal length of 10 cm.
Hence, many optical designs can be implemented with a
single lens. On the other hand the index of refraction n for
X-rays in matter reads (James, 1967)
n ˆ 1 ÿ ÿ i;
…2†
ˆ …NA =2†r0 2 …Z ‡ f 0 †=A;
…3†
ˆ =4:
…4†
with
Here, NA is Avogadro's number, r0 is the classical electron
radius, ˆ 2=k1 ˆ 2c=! is the photon wavelength,
Z ‡ f 0 is the real part of the atomic scattering factor
including the dispersion correction f 0 , A is the atomic mass,
is the linear coef®cient of attenuation and is the density
of the lens material. Firstly, the real part of n is smaller than
1. A focusing lens must have a concave form rather than a
convex one. Secondly, 1 ÿ is very close to 1, being of the
order of 10ÿ6. Thirdly, although is a number small
compared with 1, X-ray absorption in the lens material is
not negligible. This has led to the widespread view that
there is no chance of manufacturing refractive X-ray lenses.
However, the problems can be overcome by stacking many
individual lenses behind one another, by making the radius
of curvature small (e.g. 0.2 mm) and by choosing a low-Z
Figure 1
Parabolic compound refractive lens (CRL). The individual lenses
(a) are stacked behind one another to form a CRL (b).
Bruno Lengeler et al.
which is short compared with the focal length. The lenses
described in this paper are made from polycrystalline
aluminium by a pressing technique and have been designed
and manufactured at the University of Technology in
Aachen. The pressing tools consist of two convex paraboloids with rotational symmetry facing each other and
guided in a centring ring. The aluminium blank in which the
paraboloids are to be pressed from both sides is held and
centred by a ring, 12 mm in diameter, that ®ts tightly into
the centring ring. The parabolic lens pro®le is simultaneously pressed into the aluminium from both sides, in a
similar way as metallic money pieces are coined. Modern
computer-controlled tooling machines allow the pressing
tool to be manufactured with a micrometre precision.
Fig. 2 shows a height pro®le through one side of a
paraboloid as measured by white-beam interferometry. The
radius of curvature at the apex of the parabola is
196 1 mm, close to the design parameter R ˆ 200 mm.
The surface roughness is about 0:1 mm r.m.s. This will be
shown to have a negligible in¯uence on the lens performance. The individual lenses are stacked behind each other
by means of two high-precision shafts so that the optical
axes are aligned with a precision of 1 mm. An assembled
CRL has the size of a matchbox.
1155
It is very easy to align the lenses in the beam. The lens
holder, shown in Fig. 3, has an alignment hole which allows
the optical axis at the CRL to be orientated parallel to the
synchrotron radiation beam. Once this alignment is
performed, the whole lens holder is translated down by
10 mm, so that the beam hits the centre of the lenses. A ®ne
adjustment is performed by means of an Si-PIN diode. The
whole alignment takes about 15 min.
In order to test the stability of an aluminium CRL in the
white beam of an undulator at the ESRF (50 W) we have
measured the temperature as a function of time in the ®rst
lens of the stack (N ˆ 25) by means of a thermocouple. The
CRL was in still air. Fig. 4 shows that the temperature
stabilizes below 423 K after 15 min. We conclude that
CRLs are able to cope without problems with the heat load
from undulators in third-generation synchrotron radiation
sources. An additional cooling can easily be applied when
the CRLs are intended to be applied in more powerful
X-ray beams.
In the following we shall calculate the imaging properties
of CRLs, including absorption and surface roughness.
Figure 3
Schematic view of a compound refractive lens.
Figure 2
Height pro®le of an aluminium lens measured by white-light
interferometry; contour plot (a) and linear scan through the lens
(b). The parabolic ®t gives R = 196 1 mm and a surface
roughness of 0.1 0.1 mm r.m.s.
Figure 4
Temperature increase in an aluminium CRL when exposed to the
white beam of an ESRF undulator (50 W). Different aluminium
absorbers in front of the CRL reduce the power load.
1156
Parabolic refractive lenses
3. Imaging of an undulator source by a CRL
First, we would like to summarize a few properties of
undulator sources installed at third-generation storage
rings. An undulator source has a ®nite size described by
Gaussians in the vertical and in the horizontal direction,
ÿ
…6†
W…x† ˆ …2v2 †ÿ1=2 exp ÿx2 =2v2 ;
where v is the r.m.s. width in the vertical direction. There is
a similar expression, W…y†, for the horizontal direction. At
the ESRF a high- undulator (considered here) has a size
v ˆ 14:9 mm and h ˆ 297 mm which corresponds to a
FWHM of dv ˆ 35 mm in the vertical and dh ˆ 700 mm in
the horizontal. The values quoted here have an uncertainty
of at least 10%.
All X-ray sources available at present, including undulators at third-generation synchrotron radiation sources,
are chaotic sources because the emission is spontaneous in
any mode, rather than stimulated in a well de®ned ®xed
mode (Loudon, 1983). An undulator harmonic has a typical
bandwidth of 1% at 10 keV. This corresponds to a longitudinal coherence time 0 of 4 10ÿ17 s. This is the time
scale over which the phase of the electric ®eld emitted by
one source point undergoes random ¯uctuations. Even if
the beam is monochromated by Bragg re¯ection to within
E=E ˆ 10ÿ4 , 0 is still only 4 10ÿ15 s. This is much
shorter than the detector response time of any detector
presently available. In other words the intensity measured
by any detector is an average over a time much longer than
the characteristic time 0. The second characteristic feature
of the chaotic source is that the phases of the electric ®eld
emitted by different source points are uncorrelated. These
properties will be used extensively in the following arguments.
Let us consider a CRL with focal length f at a distance L1
from an undulator. In general the beam passes through a
double-crystal monochromator. The monochromatic
X-rays should have energy E ˆ h- ! with a bandwidth
Figure 5
Imaging an undulator source S by a CRL (top) and system of
coordinates used in the text (bottom).
! ˆ 2 much smaller than !. The X-rays are assumed
to be linearly polarized in the horizontal plane. A typical
high- undulator at the ESRF, e.g. ID22, has a beam
divergence of 25 mrad FWHM. At a distance of 40 m it
illuminates homogeneously an object (like a lens) of 1 mm
diameter. We ask for the ®eld amplitude E…t† at a given
time t at a point P behind the lens (Fig. 5). If we place a
detector at P, the number of photons detected during a
data-collection time T will be proportional to
I ˆ …c=4†hjE…t†j2 i ˆ …c=4T†
T=2
R
ÿT=2
dt jE…t†j2 :
…7†
The second equality de®nes the detector average. As
mentioned above, the detector collection time is always
much longer than the characteristic ¯uctuation time 0 in
the phase of the electric ®eld E…t†. The ®eld at P is the
superposition of the amplitudes emitted by the individual
source points and propagating via all possible trajectories
which pass through the lens. One of these trajectories with
an optical path is shown in Fig. 6.
The ®eld amplitudes in the source which contribute to
the amplitude at P at time t have to be taken at the retarded
time t ÿ =c. Therefore, each amplitude will contain a
factor
exp‰ÿi!…t ÿ =c†Š ˆ exp…ÿi!t† exp…ik1 †;
…8†
with ! ˆ k1 c. Part of the time the photons will propagate in
free space (or in air) and part of the time they propagate
through the lens material. We denote by t12 the amplitude
for transmission at an air±lens interface and by t21 that at a
lens±air interface. So, the amplitude at P at time t for one
trajectory is proportional to (Fig. 6)
PS ˆ exp…ÿi!t† exp…ik1 Ui S†…L †N exp…ik1 PU0 †:
…9†
The amplitude L for propagation through one lens is
given by
L ˆ exp 2ik1 ‰w0 ‡ …n ÿ 1†wŠ t12 t21 :
…10†
The layer of thickness d (Fig. 1) generates a phase shift
which is independent of u, v and is therefore discarded
here. However, it is taken into account in the transmission
of the lens. In Appendix B it is shown that the transmission
amplitude …t12 t21 †N for N lenses is given by
…t12 t21 †N ˆ exp ÿNQ20 2 s2 =R2 ;
…11†
Figure 6
Trajectory for a photon emitted in S and propagating to P via the
lens.
Bruno Lengeler et al.
Q0 ˆ k1 ;
ˆ exp
ÿNQ20 …12†
2
;
…13†
where is the r.m.s. roughness of a lens surface and Q0 is
the momentum transfer for transmission through an air±
lens interface at normal incidence. Since the roughnesses of
individual lenses are not correlated, the effective roughness
for N lenses with 2N surfaces is …2N†1=2 . Since refraction is
very small and since
2R0 ; 2Nw0 L1 ; L2 ;
…14†
we have assumed that the trajectory in the lens is parallel to
the optical axis. Hence,
ÿ ÿ
PS ˆ exp…ÿi!t† exp ik1 Ui S ‡ PU0
…15†
‡ 2N w0 ‡ …n ÿ 1†w …t12 t21 †N :
Expansion of Ui S to quadratic terms in xj , yk , u, v gives
Ui S ˆ ‰…u ÿ xj †2 ‡ …v ÿ yk †2 ‡ …L1 ÿ Nw0 †2 Š1=2
…16†
The next term in the expansion of the root gives a contribution to the phase
2 …17†
k1 …u ÿ xj †2 ‡ …v ÿ yk †2 = 8…L1 ÿ Nw0 †3 :
Now, u, v, xj , yk are not larger than 500 mm and L1 is at least
Ê photons the phase corresponding
1 m (often 40 m). For 1 A
to expression (17) is smaller than 2 10ÿ4 rad. For this
reason higher-order terms in (16) have been neglected.
Inserting (17) and the corresponding one for PU0 in (15)
we obtain
…18†
PS ˆ Kj exp ÿs2 ÿ ik1 Gs cos…' ÿ '1 † ;
‡ …x2j ‡ y2k †=2L1 ‡ …p2 ‡ q2 †=2L2 ;
…19†
2R2 ˆ NR ‡ 2NQ20 2 ÿ ik1 FR2 a ÿ ibF;
…20†
with
u ˆ s cos ';
xj =L1 ‡ p=L2 ˆ G cos '1 ;
b ˆ k1 R 2 ;
v ˆ s sin ';
…21†
yk =L1 ‡ q=L2 ˆ G sin '1 ;
…22†
F ˆ 1=L1 ‡ 1=L2 ÿ 1=f ;
…23†
G2 ˆ …xj =L1 ‡ p=L2 †2 ‡ …yk =L1 ‡ q=L2 †2 :
…24†
Summing over all trajectories through the lens results in an
integral over ' from 0 to 2 and over s from 0 to R0, giving
PS ˆ Kj ÿ;
RR0
0
ds 2s exp…ÿs2 †J0 …k1 Gs†:
…26†
Before solving the integral we will consider a few interesting special cases.
3.1. Case 1: large transparent CRL
If absorption and roughness are zero ( ˆ 0 and ˆ 0),
the integral equation (26) reads
ÿ ˆ 2
RR0
0
ds s exp…ik1 Fs2 =2†J0 …k1 Gs†;
…27†
and can be solved by the method of stationary phase
(Jackson, 1975). The stationary points of the phase are at
s ˆ 0 and F ˆ 0. The point s ˆ 0 does not contribute to the
integral because sJ0 …k1 Gs† ˆ 0. Therefore F ˆ 0 is the
relevant solution. In other words we expect everywhere
destructive interference, except for F ˆ 0,
RR0
0
ds s J0 …k1 Gs† ˆ 2R0 J1 …k1 GR0 †:
…28†
The Bessel function J1 of order 1 ¯uctuates strongly if R0 is
large. When averaged over a ®nite detector size the
amplitude vanishes, unless G ˆ 0, in which case ÿ ˆ R0 2.
We conclude that if R0 is large, if absorption and roughness
are zero, we expect a point to point imaging by the lens,
given by the intersection of the plane F ˆ 0 at a distance L2
equal to
L20 ˆ fL1 =…L1 ÿ f †;
…29†
and the ray G ˆ 0, i.e.
xj =L1 ‡ p=L2 ˆ 0;
yk =L1 ‡ q=L2 ˆ 0:
…30†
This is a result well known from optics textbooks.
3.2. Case 2: very small aperture 2R0 (pinhole)
Kj ˆ exp ÿi!t ‡ ik1 L1 ‡ L2
a ˆ NR ‡ 2NQ20 2 ;
ÿˆ
ÿ ˆ 2
ˆ L1 ÿ Nw0
‡ ‰…x2j ‡ y2k † ‡ …u2 ‡ v2 † ÿ 2…uxj ‡ vyk †Š=2L1 :
1157
…25†
When R0 is very small, the effect of absorption and
roughness is negligible and the phase factor in (26) can be
expanded to terms linear in s2 giving
ÿ ˆ 2
RR0
0
ds s…1 ‡ ik1 Fs2 =2†J0 …k1 Gs†:
…31†
The oscillating Bessel function in combination with a ®nite
detector size makes the integral vanish, unless G ˆ 0. In
other words a pinhole generates intensity everywhere on
the ray G ˆ 0 (point-to-line imaging), resulting in a very
large depth of ®eld. The expansion of the phase factor is
justi®ed if R0 is much smaller than the radius of the ®rst
Fresnel zone. The same point-to-line imaging is produced if
the absorption in the lens is so large that the effective
aperture becomes small. We will now show that real CRLs
have a behaviour between the two extreme cases of pointto-point and point-to-line imaging. It turns out that a large
depth of ®eld is a characteristic feature of refractive X-ray
lenses. We have found no way to solve equation (26)
analytically. However, for most practical cases, R0 is chosen
so large that
1158
Parabolic refractive lenses
ap aR20 =2R2 1:
…32†
Then, the upper limit of the integral can be replaced by
in®nity and the integral can be solved analytically (Abramowitz & Stegun, 1972),
ÿ
…33†
PS ˆ Kj …2=2† exp ÿk21 G2 =4 :
This is (apart from an irrelevant proportionality constant)
the ®eld amplitude in P at time t generated by a monochromatic point source in …xj ; yk † and transmitted through
all possible trajectories through the CRL.
We now have to consider the effect of the ®nite bandwidth of the quasi-monochromatic light. According to
expression (8), the ®eld amplitude in P at time t is now
proportional to
R
d! g…!† exp‰ÿi!…t ÿ =c†Š;
…34†
where g…!† is the frequency distribution of width ! for
the quasi-monochromatic light. The intensity I measured
by a detector at P [see equation (7)] is proportional to
R
hjHj2 i d! d!0 g…!†g …!0 † exp‰i…! ÿ !0 †=cŠ
…1=T†
T=2
R
ÿT=2
dt exp‰ÿi…! ÿ ! †tŠ :
0
P
j
ÿ
exp ik1 x2j =2L1 exp‰iv …xj t†Š
exp
hÿ
ÿ
2 i
ÿ k21 =4jj2 xj =L1 ‡ p=L2 ;
…35†
In other words, when the detector collection time is much
longer than the characteristic ¯uctuation time 0 then it is
the intensities (and not amplitudes) for different frequencies which have to be added. Therefore, for the time being,
we consider a monochromatic ®eld E…t† and include the
effect of the ®nite frequency bandwidth later, if needed.
Finally, we have to include the effect of the extended
chaotic source with the dimensions quoted in equation (6).
This feature is taken into account by including in equation
(33) a phase factor exp‰iv …xj t† ‡ ih …yk t†Š and summing
over the different emitters …j; k† in the source. The phases
v …xj t† and h …yk t† ¯uctuate statistically from source point
to source point since the individual emitters in a chaotic
source are independent of each other. The ®eld amplitude
at the observation point P at time t reads
…37†
…38†
and similarly for ÿh . The intensity measured by the detector
is again proportional to the modulus squared of ÿv and ÿh ,
averaged over the detector collection time T,
P
hjÿv j2 i ˆ h…xj †h …xj0 †
jj0
hexpfi‰v …xj t† ÿ v …xj0 t†Šgi;
…39†
where h…xj † denotes the remaining factors in the integrand
of (38). For a given pair j; j0 of emitter coordinates the
phase difference in the exponent of (39) ¯uctuates within
the range 0±2 in the time scale 0. If the detector collection time T is much longer than 0 the contribution
averages to zero, unless j ˆ j0,
hexpfi‰v …xj t† ÿ v …xj0 t†Šgi ˆ jj0 :
…40†
In other words, for a chaotic X-ray source, and for T 0 ,
the contribution of the individual source points have to be
added incoherently,
n
ÿ
2 o
P
hjÿv j2 i ˆ exp ‰ÿk21 <…†=2jj2 Š xj =L1 ‡ p=L2
j
The complex conjugate of a quantity is marked by an
asterix. As mentioned above, even for a beam monochromated to within E=E ˆ 10ÿ4 the characteristic ¯uctuation time 0 is 4 10ÿ15 s, which is much shorter than
the fastest detector response. So, the phase …! ÿ !0 †T is
approximately 2T=0 which is much larger than 1 and the
time integral in equation (35) can be replaced by
2…! ÿ !0 †, giving
R
…36†
hjHj2 i ˆ …2=T† d!jg…!†j2 jPS j2 :
P …p; q† ˆ …2=2† exp…ÿi!t† exp‰ik1 …L1 ‡ L2 †Š
exp ik1 …p2 ‡ q2 †=2L2 ÿv ÿh ;
ÿv ˆ
R
dx W…x† exp ‰ÿk21 <…†=2jj2 Š
ÿ
2
xj =L1 ‡ p=L2 :
ˆ
…41†
In the last line we have replaced the sum over the individual emitters by an integral with the Gaussian distribution
W…x† given by equation (6). The integral can be solved
analytically. Including the similar contribution for hjÿh j2 i
we obtain for the intensity at P,
I…p; q† hjP j2 i
ˆ 2 2 =‰…aav ‡ b2 F 2 †…aah ‡ b2 F 2 †Š1=2
exp …ÿak21 R2 =L22 † p2 =…aav ‡ b2 F 2 †
‡ q2 =…aah ‡ b2 F 2 † ;
av ˆ a ‡ 2k21 R2 v2 =L21 ;
ah ˆ a ‡ 2k21 R2 h2 =L21 :
…42†
…43†
A word should be said about the loss of observability of
interference for the ®eld amplitudes emitted by different
source points in a chaotic source. It is not exclusively a
property of the source, but also a consequence of the slow
speed of the detectors (presently) available and of the long
detector collection time chosen (long compared with 0 ). In
a similar way it was the ®nite detector spatial resolution
which was responsible for the non-observability of intensity
outside the ray G ˆ 0 in (28) and (31).
Equation (42) is the central expression for the imaging of
an undulator source by a CRL. We will now discuss the
intensity distribution at the point P with coordinates
…p; q; L2 †.
Bruno Lengeler et al.
1159
3.2.1. Intensity in the focal plane F = 0. The focal plane F =
0 …1=L1 ‡ 1=L20 ˆ 1=f † is at a distance L20 ˆ L1 f =…L1 ÿ f †
from the middle of the lens. In this plane the intensity is
distributed according to a Gaussian with width (FWHM)
expressed in terms of the mass absorption coef®cient =
and of the focal length f as
Bv ˆ 2:355 L20 …v2 =L21 ‡ a=2k21 R2 †1=2 ;
For a given focal length f and a given photon wavelength the transmission of the lens increases with decreasing mass
absorption coef®cient, i.e. low-Z elements as lens material,
like Li, Be, B, C and even Al are favourable. In Fig. 7 are
shown the mass absorption coef®cients of these elements
and of Ni. The values = ®rst drop with photon energy, as
Eÿn with n close to 3. However, at 0.2 cm2 gÿ1 the strong
decrease merges into a much slower decrease when
Compton scattering exceeds photoabsorption. Comptonscattered photons do not contribute to the image formation. Those which are scattered under a large angle are
ultimately destroyed by absorption whereas those which
are scattered in the forward direction will contribute to a
blur of the focal spot. It turns out that Compton scattering
ultimately limits the transmission and the resolution of a
CRL. The gain of a CRL is de®ned as the ratio of the
intensity in the focal spot and the intensity behind a pinhole
of size equal to the spot of the lens. It is (Lengeler et al.,
1998)
…44†
and a similar expression Bh for the horizontal, with v being
replaced by h. There are two contributions to Bv. The ®rst
one is due to the demagni®cation of the source according to
geometrical optics and is given by (44) for a ˆ 0 (i.e. for
ˆ 0 and ˆ 0). It has the value 2:355 v L20 =L1. The
other term including a [equation (20)] describes the in¯uence of diffraction by the effective aperture of the lens.
3.2.2. Effective lens aperture Deff and in¯uence of lens
surface roughness. The effective lens aperture can be
deduced from the integral equation (26). For F ˆ 0 and
G ˆ 0, the integral has the meaning of an effective lens
area. When and are zero …a ˆ 0† the integral has the
value R20. With non-vanishing values of and the
integral is equal to
…=4†D2eff ˆ R20 ‰1 ÿ exp…ÿap †Š=ap ;
…45†
giving, for the effective aperture Deff ,
Deff ˆ 2R0 f‰1 ÿ exp…ÿap †Š=ap g1=2 ;
ap ˆ
aR20 =2R2 :
…46†
…47†
For visible light and glass lenses in which absorption and
roughness are negligible (ap ˆ 0), the effective aperture
Deff is identical to the geometric aperture 2R0. Not so for
refractive X-ray lenses: here attenuation of the X-rays in
the lens material is the limiting factor for the effective
aperture. Another comment is appropriate about the
in¯uence of roughness on the performance of X-ray
mirrors as opposed to X-ray lenses. Roughness reduces the
re¯ectivity and the transmission according to an exponential factor exp(ÿ2Q2 2 ). For a mirror, the momentum
transfer Q is 2k1 with a typical value of 0:1 for the angle
Ê ÿ1. For an
Ê , Q = 2.2 10ÿ2 A
of re¯ection . For = 1 A
aluminium CRL at 12.4 keV with = 3:54 10ÿ6 , f = 1 m,
N = 28, the momentum transfer is …2N†1=2 k1 =
Ê ÿ1. The low value of Q0 for a CRL allows for
1:7 10ÿ4 A
much larger values of . If a mirror for hard X-rays needs a
surface ®nish of 1 nm r.m.s., a CRL needs a ®nish of 1 mm.
This is a dramatic difference which simpli®es drastically the
requirements for manufacturing refractive X-ray lenses.
3.2.3. Transmission and gain of a CRL. The transmission
of a CRL with N individual lenses and a thickness d
between the apices of the parabolas (Fig. 1) is
Tp ˆ exp…ÿNd†…1=R20 †
RR0
0
2ap ˆ …=†…R20 =f †fA=‰Na r0 2 …Z ‡ f 0 †Šg:
gp ˆ Tp …4R20 =Bv Bh †;
…49†
…50†
where Bv and Bh are the FWHM size of the focal spot in the
vertical and horizontal directions.
3.2.4. Comparison with experimental data. We have
imaged the ID22 source at ESRF with an aluminium CRL
Ê ) photons. The lens parameters are N =
and 15 keV (0.83 A
33, R = 200 mm, 2R0 = 0.87 mm, = 0.1 0.1 mm, =
2:41 10ÿ6 , = 20.52 cmÿ1, L1 = 63.00 0.05 m. The
values for R and 2R0 are the same throughout the whole
article. For this set of parameters, NR + 2Nk12 22 = a =
13.54 0.47 + 0.02 0.03 = 13.58 0.50, ap = 32.12 5.74,
f = 1.26 0.04 m, L20 = 1.29 0.04 m, Deff = 154 10 mm.
It is obvious that the roughness of 0.1 mm in the lens surface
gives a minor contribution to ap and hence to the effective
ds 2s exp…ÿas2 =R2 †
ˆ exp…ÿNd†…1=2ap †‰1 ÿ exp…ÿ2ap †Š:
…48†
For good quality lenses the in¯uence of surface roughness
is a minor contribution to a (see below). Then ap can be
Figure 7
Mass attenuation coef®cient = for Li, Be, B, C, Al and Ni. For
Be and Al the photoabsorption coef®cent = is also shown (thin
lines) (Henke et al., 1993).
1160
Parabolic refractive lenses
lens aperture. With 2k21 R2 ˆ …4:58 0:23† 1014 , v =
14.9 mm and h = 297 mm we calculate a spot size
Bv ˆ 0:9 0:1 mm in the vertical and Bh ˆ 14:2 1:4 mm
in the horizontal. The intensity was measured with a
detector which had a point spread function of 1:2 0:1 mm
FWHM (Koch et al., 1998). So, the expected overall spot
size is 1:5 0:1 mm in the vertical and 14:3 1:4 mm in the
horizontal. Fig. 8 shows the experimental result with a
horizontal and a vertical scan. The measured spot size is
1:6 0:3 mm 14:1 0:3 mm, in good agreement with the
expected result. Note that the rugged structure in the
horizontal line scan is not noise but re¯ects structure in the
incoming intensity, which we believe is due to the effect of
windows, absorbers and the monochromator crystals.
The transmission of the lens considered here
(d ˆ 16 2 mm) is Tp ˆ 0:53 0:16%. The theoretical
gain of the lens is 200 40 compared with an experimental
value of 177 35. The great advantage of beryllium over
aluminium becomes visible in the transmission, e.g. a
beryllium lens with N ˆ 52 has f ˆ 1:28 m at 15 keV. When
all geometrical parameters are kept the same, then
ˆ 0:52 cmÿ1, a ˆ 0:64 and ap ˆ 1:60. The effective
aperture now becomes 633 mm and the transmission is 30%,
an increase by a factor of 60 compared with aluminium. The
gain compared with a pinhole of equal size would be 11 800.
3.2.5. Intensity on the optical axis G = 0. On the optical
axis the intensity is distributed over a much larger range in
space. The longitudinal distribution on the ray G ˆ 0 has,
according to equation (42), a FWHM Bl given by
Bl ˆ ‰L220 …2a†1=2 =k1 R2 Š
‰…a2v ‡ a2h ‡ 14av ah †1=2 ÿ …av ‡ ah †Š1=2 ;
…51†
with av and ah given in equation (43).
In the case of the aluminium lens described in the
previous section, Bl = 44.8 mm. This is a very large depth of
®eld, which is typical for refractive X-ray lenses and which
is mainly due to the small effective aperture of the lens.
3.2.6. Laterally coherent secondary source. Coherent
X-ray scattering is a new spectroscopy for studying dynamical processes, which extends speckle spectroscopy with
laser light from the mm length scale to the nm range
(Brauer et al., 1995; Thurn-Albrecht et al., 1996; Mochrie et
al., 1997). The spectroscopy needs a laterally coherent
X-ray beam. The CRLs are excellently suited to generate a
secondary source with this property if they are installed at
an undulator at third-generation synchrotron radiation
sources. According to Fig. 9 the CRL generates an image of
the source which illuminates an area of dimension dill on
the sample,
dill ˆ Deff L3 =L2 :
…52†
On the other hand, the lateral coherence length of the
secondary source at the position of the sample is
ltr ˆ L3 =Bdem
v ;
…53†
where Bdem
dv L2 =L1 is the size of the secondary source
v
given by the demagni®cation by the lens. The sample is
coherently illuminated if dill ltr, i.e. if
…=Deff †L2 Bv diff :
Bdem
v
…54†
In other words, when diffraction limits the size of the
secondary source, then this source can be considered as a
point source which illuminates the sample coherently in the
lateral direction. Equation (54) may also be written as
Deff L1 =dv :
…55†
Ê , L1 = 63 m and dv = 2.355 v = 35 mm, this gives
For = 1 A
Deff 180 mm. In the vertical direction this condition is
ful®lled for the aluminium CRL described in this paper. In
the horizontal plane, however, the secondary source is not
diffraction limited. Additional means have to be found to
Figure 8
ID22 undulator source of the ESRF imaged by an aluminium CRL
with 15 keV photon energy (a) and linear scans in the horizontal
and vertical directions resulting in a 14.1 1.6 mm2 spot size
(FWHM) (b).
Figure 9
Set-up for generating a laterally coherent secondary X-ray source,
which illuminates an object coherently.
Bruno Lengeler et al.
reduce the effective source in the horizontal in order to
illuminate the sample coherently in both directions.
3.2.7. Highly parallel X-ray beam generated by CRL. The
beam emitted by an undulator has already a small divergence of typically 25 mrad and generates a spot size of
1 mm at 40 m from the source. By means of a refractive
lens the beam can be made even less divergent, when the
source is in the focal spot of the lens. For a single aluminium lens (N ˆ 1) at 15 keV with ˆ 2:41 10ÿ6 the focal
length is 41.43 m. This is a typical value for the distance
between the undulator and the ®rst experimental hutch at
ESRF beamlines. For a single beryllium lens (N ˆ 1) with
ˆ 2:37 10ÿ6 at 12 keV the focal length is 42.24 m. In
both cases R was assumed to be 0.2 mm. When N ˆ 1, the
absorption parameter ap ˆ aR20 =2R2 is small (e.g. it is 1.03
for aluminium at 15 keV), and it is no longer appropriate to
replace the upper integration boundary in equation (26) by
in®nity. It is somewhat tedious to solve the integrals for
calculating the intensity distribution. However, the main
results on the beam divergence are evident in the light of
the results from the previous sections. When the source is in
the focal spot of a single lens, the beam divergence is
v ˆ ‰…dv =L1 †2 ‡ …1:029=Deff †2 Š1=2 ;
…56†
and a corresponding expression exists for h . The ®rst
term is due to the ®nite source size and the second is due to
diffraction at the effective aperture Deff given by equation
(46). A single aluminium lens at 15 keV with a = 0.41, f = L1
= 41.43 m, Deff = 708 mm, has a divergence v = 0.85 mrad
which is at least a factor of 20 smaller than the divergence
of the undulator without lens. In the horizontal the
reduction in divergence is less dramatic with h =
16.8 mrad. In both cases it is the source size which determines the beam divergence behind the lens.
3.2.8. Imaging of an X-ray free-electron laser source by a
CRL. There are plans to build an X-ray free-electron laser
(XFEL) based on the self-ampli®ed-spontaneous-emission
(SASE) principle (Ingelman & Jonsson, 1997). It is hoped
that the laser will emit a laterally coherent X-ray beam in
the TEM00 mode, which is a Gaussian delimited spherical
wave. The divergence is expected to be 1 mrad. At a
distance of 1000 m an object of 1 mm in diameter is laterally coherently illuminated. A typical wavelength of this
Ê (Schneider, 1997). Under these condisource will be 1 A
tions the source can be considered as a point source
…v ˆ h ˆ 0†, although the geometrical source is 50 mm in
diameter. According to equation (44) the image of the
source will have a transverse FWHM Btr of
Btr ˆ 0:75…L20 =2R†…a=2†1=2 ˆ 0:75…L20 =Deff †;
1161
4. Imaging of an object illuminated by an undulator
beam by means of a CRL
4.1. Theory of imaging
In this section we consider the concept of a microscope
working in the hard X-ray regime between 5 and 60 keV.
Fig. 10 shows a schematic set-up. The object, illuminated by
the undulator, is characterized by a transmission function
T…x; y†. For a thin slab of thickness D in the beam direction
with index of refraction n…x; y; z†, T…x; y† reads
D
R
…58†
T…x; y† ˆ exp ik dz n…x; y; z† ÿ 1 :
0
The object is located slightly outside of the focal length f of
the CRL …L1 >f †. Then the lens images the object in the
image plane at a distance L20 ˆ f L1 =…L1 ÿ f † on a highresolution detector. The magni®cation of the image is
1=m ˆ L20 =L1 :
…59†
At the beamline ID22, L20 may be chosen as large as 25 m.
For a focal length of 1 m this results in a magni®cation of at
least 20. The main advantage of this magni®cation is in the
relaxed requirements on the detector. We will see that the
resolving power of the microscope may be as good as
0.1 mm. However, there is no detector available with this
resolution. Our high-resolution detector (Koch et al., 1998)
has a point spread function of 1 mm FWHM and is, owing
to the magni®cation, able to resolve details in the object
below 0.1 mm in size.
We now calculate the intensity distribution in the
detector, in a similar way as performed in x3. The divergence of the X-ray beam emitted by the undulator is about
25 mrad. The distance L to the object is typically 42 m at the
ESRF. As a consequence, every source point illuminates
every object point when the object is below 1 mm in
diameter. The ®eld amplitude at the exit of the illuminated
object in the point (x, y) is
…x; y† ˆ exp‰ÿi…!t ÿ k1 L†ŠT…x; y†
P
exp i‰…Xj ÿ x†2 =2LŠ ‡ iv …Xj t†
jk
‡ i‰…Yk ÿ y†2 =2LŠ ‡ ih …Yk t† :
…60†
This amplitude has to be transferred through the lens to the
observation point P in the detector plane, in a similar way
as performed in x3. The amplitude at the point P at time t is
…57†
given by the diffraction at the effective aperture of the lens.
For a beryllium CRL with N = 45 individual lenses at
12.4 keV ( = 2.22 10ÿ6, = 0.70 cmÿ1, = 0.2 mm, d =
20 mm) we obtain a focal length of f = 1 m. The parameter
ap is 1.75. This results in a transmission of 26%, an effective
aperture of Deff = 614 mm, a focal spot size of Btr = 0.12 mm
and a gain of 107.
Figure 10
Set-up for a hard X-ray microscope with coordinate system used
in the text.
1162
Parabolic refractive lenses
…p; q; t† ˆ …=† exp ÿi!t ‡ ik1 ‰L ‡ L1 ‡ L2
P
Njk ;
‡…p2 ‡ q2 †=2L2 Š
jk
Njk ˆ
R
…61†
dx dy Kv …Xj ; x; t†Kh …Yk ; y; t†T…x; y†;
with
Kv ˆ exp iv …Xj ; t† ‡ ik1 …Xj ÿ x†2 =2L ‡ x2 =2L1
ÿ …k21 =4jj2 †…x=L1 ‡ p=L2 †2 ;
…62†
and similarly for Kh .
The intensity measured in the detector is proportional to
hjp …pqt†j2 i with the detector average given by equation
(7). This expression contains the integral
…1=T†
T=2
R
ÿT=2
dt exp‰ÿiv …Xj t† ÿ iv …Xj0 t†Š ˆ …2=T†jj0 :
…63†
As in x3, the Kronecker delta is a consequence of the long
detection time T 0. A similar Kronecker delta is
obtained for the summation over k and k0 . The summation
over the source points j (and k) is replaced by integration
over a Gaussian distribution, giving
P
exp ÿik1 ‰…x ÿ x0 †=LŠXj
j
R1
dX exp…ÿX 2 =2v2 †
ˆ …2v2 †ÿ1=2
ÿ1
exp ÿik1 ‰X…x ÿ x0 †=LŠ
ˆ exp ÿ…x ÿ x0 †2 =2lv2 ;
where
lv ˆ L=2v ˆ L= = …2 ln 2†1=2 dv
…64†
…65†
is the transverse coherence length in the x-direction. A
similar value is de®ned for the y-direction. Two points x and
x0 in the object are illuminated coherently by the source if
jx ÿ x0 j lv . In that case the ®eld amplitudes emitted by
these two points have a ®xed phase relation. We thus obtain
for the intensity,
microscope and by reconstructing the three-dimensional
distribution n…x; y; z† in equation (58) via a tomographic
approach. Techniques have been developed for how this
may be performed (Gilboy, 1995; Grodzins, 1983; Snigirev
et al., 1995; Raven et al., 1996, 1997; Spanne et al., 1999;
Momose, 1995; Beckmann et al., 1997). Here, we would like
to consider the transversal and longitudinal resolving
power of our microscope. For that purpose we assume that
the object is a non-transparent screen with two pinholes at
the positions …x0 ; 0† and …ÿx0 ; 0† along the x-axis,
T…x; y† ˆ ‰…x ÿ x0 † ‡ …x ‡ x0 †Š…y†:
…68†
The integrals in equation (66) can now be solved analytically, giving
2
…2R2 †2 2
2
2 2 2
2 2 q
exp ÿk1 R =a ‡ b F a 2
hjP …p; q†j i ˆ 2
a ‡ b2 F 2
L
2
2
k21 R2
x0 p2
a
‡
2 exp ÿ 2
a ‡ b2 F 2 L21 L22
k21 R2
2x0 a
p
cosh 2
a ‡ b2 F 2 L 1 L 2
…2x †2
‡ exp ÿ 02
2lv
k2 R2 2x bF
…69†
cos 2 1 2 2 0 p :
a ‡ b F L1 L2
We will ®rst consider two extreme regimes, that of incoherent and that of coherent illumination of the object.
4.1.1. Incoherent illumination: 2x0 lv . If the lateral
coherence length is very small …2x0 lv †, then the second
term in the wavy brackets [equation (69)] is negligible and
the intensity in the focal plane F = 0 is proportional to
hjP …p†j2 i ' exp …ÿk21 R2 =a†… p=L2 0 ‡ x0 =L1 †2
‡ exp …ÿk21 R2 =a†… p=L20 ÿ x0 =L1 †2 : …70†
The intensity is given by two Gaussians (Fig. 11) at a
distance 2x0 L20 =L1. We de®ne as transverse resolving
…jj2 =2 2 †hjP …p; q†j2 i ˆ
R
^ h …y; y0 †T…x; y†T …x0 ; y0 †; …66†
^ v …x; x0 †K
dx dx0 dy dy0 K
with
^ v …x; x0 † ˆ exp ÿ…x ÿ x0 †2 =2lv2
K
exp i…k1 =2†…1=L ‡ 1=L1 †…x2 ÿ x02 †
exp …ÿk21 =4jj2 † …x=L1 ‡ p=L2 †2
2
0
‡ …x =L1 ‡ p=L2 † ;
…67†
^ h.
and the corresponding expression for K
We now have to consider the object more in detail. If the
object is of unknown structure, we may determine its
structure by imaging it by means of our hard X-ray
Figure 11
Intensity distribution for two object points at distance 2x0
illuminated incoherently (a) and coherently (b).
Bruno Lengeler et al.
power dt that distance 2x0 in the object for which the image
points are separated by the FWHM of each image. This
gives
1=2
dinc
t ˆ ‰2…2 ln 2† =Š…L1 =Deff †
 0:75…L1 =Deff † ˆ 0:75…=2NA†;
…71†
where NA is the numerical aperture,
NA ˆ Deff =2L1 :
…72†
In the present calculation we have assumed R0 to be very
large, then Deff ˆ 2R…2=a†1=2 . When ap is not large, then
Deff is given by equation (46). The intensity at p ˆ 0 is
93.7% of the value at the maxima (Fig. 11). For the case of
the aluminium CRL at 15 keV mentioned in x3.2.4 (N = 33,
f = 1.26 m, ap = 32.12 and Deff = 154 mm) and an object-tolens distance of L1 = 1.36 m, the resolving power is
0:55 0:01 mm and the numerical aperture is NA =
95:66 0:08 10ÿ5 . It is the low NA (`slender' X-ray
optics) which is responsible for the resolution being much
larger than the photon wavelength. Nevertheless, a value of
dt = 0.14 mm can be achieved with an aluminium CRL at
40 keV when the transmission in aluminium is improved
compared with the value at 15 keV.
The low value of the NA results in a very poor longitudinal resolving power dl of the hard X-ray microscope.
Indeed, for p = q = 0 and x0 = 0,
hjP …L2 †j2 i ˆ …2R2 †2 2 =…a2 ‡ b2 F 2 †:
…73†
The longitudinal resolving power dl is de®ned as the
distance of two object points along the optical axis whose
image distance is as large as the spread of one image point
along the optical axis.
dl ˆ 2…a=b†L21 ˆ …8=†…L21 =D2eff † ˆ …2=†…1=…NA†2 :
…74†
In the case of the aluminium CRL with N ˆ 33 at 15 keV,
dl ˆ 16:5 0:1 mm. Note that dt is proportional to the
inverse NA whereas dl is proportional to the square of the
inverse NA as is well known in optics (Lipson et al., 1998).
4.1.2. Coherent illumination: 2x0 lv . If the lateral
coherence length is large …2x0 lv †, then the second term
in the wavy brackets [equation (69)] is 1 for F ˆ 0 and the
intensity in the focal plane is proportional to
( "
2 #
k21 R2 p
x0
2
ÿ
hjP …p†j i ˆ exp ÿ
2a L20 L1
"
2 #)2
k21 R2 p
x0
‡
: …75†
‡ exp ÿ
2a L20 L1
Fig. 11 shows the intensity distribution for coherent and
incoherent illumination. If the resolving power is de®ned
again by the separation of the amplitude peaks being equal
to the FWHM of an amplitude peak we ®nd
ˆ 21=2 dinc
dcoh
t
t ˆ 1:06L1 =Deff ˆ 1:06…=2NA†:
…76†
1163
In that case the indentation at p ˆ 0 is 14.0%. It is well
known from optics that incoherent illumination improves
the lateral resolution (Lipson et al., 1998).
4.2. Comparison with the experiment
We have imaged a few objects with the new CRL. The
object is illuminated from behind by the X-ray source and
imaged through the lens onto a position-sensitive detector.
Due to the low absorption of hard X-rays in air there is no
need for a sample chamber nor for evacuated beam pipes.
However, the latter might be useful in order to reduce the
background due to Compton scattering in air. In Fig. 12(a)
is shown the X-ray microscopical image of a double gold
mesh. It consists of a square mesh with a period of 15 mm, a
bar width of 3 mm and a thickness of 2 mm. Attached from
behind in one part of this mesh is a second linear mesh with
a period of 1 mm, a width of 0.5 mm and a thickness of
0.5 mm.
Ê ) photons
The mesh was imaged with 23.5 keV (0.53 A
from ID22. The aluminium lens and imaging parameters
were N = 62, f = 1:65 0:04 m, L1 = 1:79 0:01 m, L20 =
21:40 0:04 m, = 5.81 cmÿ1, Deff = 211 9 mm. An area
of 300 mm in diameter was imaged with a magni®cation of
12 0:1 and a theoretical lateral resolving power of dt =
0.34 0.02 mm. Note that the image shows almost no
distortion over the ®eld of view. Fig. 12(b) shows an
enlargement of the region inside the rectangle of Fig. 12(a).
For comparison, part of the mesh has been imaged with a
scanning electron microscope. Note that the wires behind
the bars of the coarse grid are visible in the X-ray microscope but not in the electron microscope. This illustrates
the imaging possibilities of opaque objects by the new hard
X-ray microscope.
For testing the resolution of the new microscope we have
imaged a Fresnel zone plate (Lai et al., 1992) with 169 zones
made of gold (1:15 mm thick on a Si3N4 substrate of 0.1 mm
thickness). The outermost zones had a width of 0.3 mm.
Fig. 13 shows an enlarged region which includes the border
of the Fresnel lens. The parameters of the lens and the
photon energy were the same as those from Fig. 12. The
outermost zones of the Fresnel lens are clearly resolved and
the experimentally achieved resolution is in good agreement with the expected resolution of 0.34 mm resulting
from diffraction at the ®nite aperture of the lens. Figs. 12
and 13 were recorded on Kodak high-resolution X-ray ®lm
which had a resolution of 1 mm. The smallest image structures were about 3 mm (which correspond to object structures of 0.25 mm at a magni®cation of 12). Thus the
resolution of the whole set-up was not limited by that of the
®lm.
4.3. Limits of the lateral resolving power
As outlined in equations (71) and (72) the lateral resolution dt is limited by the small value of the numerical
aperture NA, which is ultimately limited by the attenuation
of the X-rays in the lens material. Fig. 14 shows the variation of dt with photon energy E for aluminium and beryl-
1164
Parabolic refractive lenses
lium CRL under the following assumptions: f = 1 m, R =
0.2 mm, 2R0 = 0.87 mm. Also shown are the number N of
individual lenses in the stack and the transmission of the
lens (with d = 10 mm for aluminium and d = 20 mm for
beryllium).
We expect a resolution better than 0.1 mm in the case of
beryllium above 15 keV. As shown in Fig. 7 it is Compton
scattering which ultimately limits the resolution and the
transmission in beryllium. Without Compton scattering we
would expect a lateral resolution of 0.05 mm for a beryllium
CRL with f = 1 m at 20 keV and a transmission above 50%.
Since Compton scattering cannot be avoided, the question
arises if there are other ways to increase the effective
Figure 13
X-ray micrograph (23.5 keV) of a Fresnel zone plate with 169 gold
zones on a Si3N4 substrate. The outermost zone of width 0.3 mm is
clearly resolved.
Figure 12
X-ray image of a square gold mesh with a period of 15 mm in both
directions. The photon energy was 23.5 keV. (a) Full ®eld of view;
(b) enlargement of (a) with the lower squares having attached to
them a second linear gold mesh with period 1 mm; (c) scanning
electron micrograph of one square from (a) and (b). Note that the
electron micrograph in contrast to the X-ray micrograph is not
able to show the linear gold grid behind the bars of the coarse gold
lattice.
Figure 14
Lateral resolution dt , number N of individual lenses and
transmission Tp of aluminium (full line) and beryllium (dashed
line) CRL with 1 m focal length versus photon energy E.
Bruno Lengeler et al.
aperture and thus the resolution. A possibility is shown in
Fig. 15. It is a parabolic refractive lens with kinoform
pro®le,
w ˆ …s2 =2R† ÿ MW;
…77†
where M is an integer. For W = 0, equation (77) is just the
parabolic pro®le discussed up to now. The discontinuity W
in the pro®le reduces the geometrical path in the lens
material. Hence, it also reduces attenuation. There is no
necessity, as in a usual Fresnel construction, to choose W in
such a way that the path difference between neighbouring
sections is . It can just as well be m where m is a large
integer, so that 2s1 ' Deff . A typical value for m is 100. The
pro®t from this choice is a low number of steps M,
combined with an increase in the effective aperture by
…M ‡ 1†1=2 . However, there is a price to be paid. First, once
W is ®xed, the photon wavelength is ®xed, which is a very
serious handicap. Secondly, the requirements on the
precision in the shape of the lens increase with m. At the
present time it is not yet clear if this approach for
increasing the effective apperture is technically feasible.
5. Conclusions
The results of this paper may be summarized as follows.
(i) Compound refractive X-ray lenses (CRLs) with
parabolic pro®le have been constructed and successfully
tested.
(ii) They can be used for hard X-rays up to 60 keV.
(iii) They have a geometric aperture of 1 mm and are
best suited for undulator beams at synchrotron radiation
storage rings.
(iv) They are stable in the white beam of an undulator.
(v) Due to their parabolic pro®le, CRLs are genuine
imaging devices, similar to glass lenses for visible light.
(vi) We have developed the theory for imaging a source
and an object illuminated by an X-ray source, including the
effects of attenuation (photoabsorption and Compton
scattering) and of surface roughness. The central expressions are equations (42) and (66).
(vii) The parabolic CRLs have been tested for imaging
up to 41 keV. There is excellent agreement between theory
and experiment. The surface ®nish of the lenses is good
enough, so that surface roughness can be neglected.
(viii) A characteristic feature of CRLs for X-rays is their
small aperture (below 500 mm) and their small numerical
aperture (NA). Nevertheless a lateral resolution of 0.3 mm
Figure 15
Parabolic refractive X-ray lens with kinoform pro®le as a
possibility to increase the effective aperture of a CRL.
1165
has been achieved and a resolution below 0.1 mm can be
expected. Compton scattering in the lens material limits the
resolution.
(ix) The transmission ranges from a few % in aluminium
CRLs up to 30% in beryllium CRLs.
(x) The main ®elds of application of refractive X-ray
lenses with parabolic pro®le are as follows.
(a) Microanalysis with a beam in the mm range for
diffraction, ¯uorescence, absorption, re¯ectometry. The
small NA of the lens guarantees a good beam collimation of
0.3 mrad. Nevertheless, the gain of the lens is at least 100
for an aluminium CRL and is expected to be a few 1000 for
a beryllium CRL. The `pink' beam of an undulator
harmonic (E=E ' 1%) is transmitted through a CRL
without loss of focusing.
(b) Imaging in absorption and phase contrast. Refractive
X-ray lenses are complementary to other microscopical
techniques. It is possible to image opaque objects which
cannot tolerate sample preparation without changing the
state of the sample. Examples are biological cells or soil
samples which change structure and functionality when the
water is removed. A hard X-ray microscope, on the other
hand, allows for in situ investigation in the natural environment.
(c) Coherent X-ray scattering. In connection with a highbrilliance X-ray source a CRL can generate a secondary
X-ray source which is diffraction limited (at least in the
vertical direction at ESRF undulators). The sample area
illuminated by the secondary source is then smaller than
the coherence area (Fig. 9). In other words, the lens works
like an aperture, without, however, reducing the intensity
as does a pinhole. We expect CRLs to be helpful optical
devices for speckle spectroscopy with hard X-rays.
APPENDIX A
Focal length of a CRL
The focal length of a CRL is calculated by means of Snell's
law, as usual for refractive lenses. The curvature of the
beam in the stack can be taken into account. However, it
turns out that this effect is small as long as the length of the
lens is short compared with the focal length. Within
corrections of order , the focal length of a parabolic CRL
with N individual lenses in the stack is found to be (Fig. 16)
ÿ
…78†
f ˆ …R=2N† 1 ÿ N 1 ÿ u21 =2R2 ;
as measured from the middle of the CRL. u1 is the height
where the incoming beam hits the ®rst lens. The correction
term (spherical aberration) is typically below 10ÿ4 or
0.1 mm for a CRL with 1 m focal length. Due to the large
depth of ®eld of refractive X-ray lenses and due to the large
distances L1, L2 and f , the correction can be neglected for
all practical purposes and a parabolic CRL can be considered as free of spherical aberration.
1166
Parabolic refractive lenses
1 varies with the height u from the optical axis in the
parabolic lens u2 ˆ 2Rw according to
APPENDIX B
Transmission amplitude t12 for a vacuum±lens
interface
We consider the transmission of an X-ray through a
vacuum±lens interface, as illustrated in Fig. 16. In the righthanded coordinate system (u0 , v0 , w0 ) the incoming and
transmitted wave vectors k1 and k2 have the components
k1 ˆ k1 …ÿ cos 1 ; 0; sin 1 †
…79†
k2 ˆ n2 k1 …ÿ cos 2 ; 0; sin 2 †:
…80†
and
Due to Snell's law,
cos 1 ˆ n2 cos 2 ;
…81†
the momentum transfer Q for transmission has only a w0
component,
Q ˆ k1 …n2 sin 2 ÿ sin 1 †:
…82†
For the low-Z materials of interest in X-ray lenses and for
hard X-rays, the absorptive correction of the index of
refraction is about three orders of magnitude smaller than
the dispersive correction (see Table 1). Hence, we neglect
the absorption in the transmission amplitude t12 and obtain
n2 sin 2 ˆ sin 1 ÿ = sin 1 ;
…83†
Q ˆ ÿk1 = sin 1 ;
…84†
ÿ6
with typically 10 . Note the very low value of the
momentum transfer. We saw in the main body of this paper
that this low value is at the origin of the insensitivity of
refractive X-ray lenses to surface roughness, a feature in
which X-ray lenses and mirrors differ drastically. The angle
1= sin2 1 ˆ 1 ‡ u2 =R2 ;
…85†
so that
Q ˆ ÿk1 …1 ‡ u2 =R2 †1=2 ˆ ÿQ0 …1 ‡ u2 =R2 †1=2 ;
…86†
Q0 ˆ k1 :
Let us consider the transmission of s-polarized X-rays
through the interface. The tangential components of the E
and H ®elds have to be continuous. This results in the
following condition for the amplitudes A2 and A1 of the
transmitted and incoming waves,
A2 =A1 ˆ t12 ˆ ‰2 sin 1 =…sin 1 ‡ n2 sin 2 †Šhexp…iQ†i:
…87†
The ®rst factor is the usual Fresnel transmission amplitude,
as explained in optics books (Lipson et al., 1998). In the
present case it is 1 with an accuracy of =2 sin2 1, which is
10ÿ6 , as can be seen from equation (83). The second
factor is due to interface roughness. In the Rayleigh model
of roughness (Stanglmeier et al., 1992) the interface is
described by a set of parallel surfaces displaced by from
the average surface and distributed around the average
surface according to a Gaussian distribution
w…† ˆ …2 2 †ÿ1=2 exp…ÿ2 =2 2 †:
…88†
Q is the phase shift of the wave transmitted through the
surface at compared with a wave transmitted at ˆ 0.
Averaging over the distribution gives
Hence,
hexp…iQ†i ˆ exp…ÿQ2 2 =2†
ˆ exp‰ÿQ20 2 …1 ‡ u2 =R2 †=2Š:
…89†
ÿ
ÿ
t12 ˆ exp ÿQ20 2 =2 exp ÿQ20 2 u2 =2R2 :
…90†
The transmission amplitude t21 for the beam leaving the
lens has the same value. Since the roughnesses on both
sides of a lens are uncorrelated, as are the roughnesses for
different lenses, the amplitude for the transmission through
2N interfaces of N lenses in a stack is
ÿ
…91†
…t12 t21 †N ˆ exp…ÿNQ20 2 † exp ÿNQ20 2 u2 =R2 :
Within the accuracy O…† ' 10ÿ6 the same result is
obtained for p-polarized X-rays.
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